Introduction

A central goal in the study of materials is to understand and quantify the relationship between the internal structure of materials and their properties. Structure-property relationships are used for designing and improving materials, or conversely, for interpreting experimental relationships in terms of microstructural features. Ideally, the aim is to construct a theory that employs general microstructural information to make accurate property predictions. A less ambitious, but more likely goal, is the provision of structure-property relations for different classes of microstructure.

Significant progress towards this goal has been made for the linear elastic properties of random porous media. Relevant reviews of the topic have been made by (Hashin 1983) and (Torquato 1991). If the pores are isolated, can be approximated by spheroids, and occupy low to moderate volume fractions, a variety of effective medium and rigorous approximations provide good predictions. These types of materials can be termed dispersions. However, many porous materials have a more interconnected or interpenetrating structure. Even at low porosities, the pores can form large clusters, while at higher porosities the pore phase can be macroscopically interconnected, giving a bi-continuous structure. Currently, no practical theory exists that is guaranteed to accurately predict the properties of random interpenetrating porous media. For example, the predictions of effective medium theories were 25% higher than data for a porous model with just 20% porosity (Roberts & Garboczi, 1999). In this paper we address this deficiency by computing empirical structure property relations for porous media with a wide variety of microstructures.

A number of theoretical formulae have been proposed that are relevant to interpenetrating porous media. For example, effective medium theories (Hashin, 1983) were developed to extend exact results for dilute inclusions to higher volume fractions. Certain microstructures were shown a posteriori (Milton, 1984) to have properties that correspond to the theories, but the physical structures are very unusual. A different class of theories is rigorously based on realistic microstructural information. These are the classic variational bounds (Milton & Phan-Thien, 1982), which only provide an upper bound for porous media, and the recent expansion of Torquato (Torquato, 1998). The microstructural information needed to evaluate the results is quite difficult to obtain, so in practice the bounds and expansion are evaluated at 'third-order'. Even with limited information, the upper bounds and expansions are thought to give good predictions for dispersions (Torquato, 1998,Torquato, 1991, §3.5). The accuracy of either class of theories is not possible to determine a priori for realistic interpenetrating porous media, so it is difficult to use the results to either improve a material or interpret experimental data. This uncertainty has limited the application of the results. Nevertheless, effective medium theories are commonly used, and the rigorous theories are attractive because of their relatively simplicity (compared to computation). Therefore, it would be extremely useful to establish the conditions and realistic microstructure types for which the theories do make accurate predictions. To address this question, on a model by model basis, we compare various well known theories to our numerical data. However, to verify a particular theory it is important to check that it predicts both isotropic elastic moduli; i.e., prediction of the Young's modulus alone is necessary but not sufficient (even though this is usually the only parameter measured!). The subtleties of the Poisson's ratio behaviour actually provide a very effective method for showing differences between the theories and demonstrating their ranges of validity. These subtleties will be described later in the paper.

The macroscopic elastic properties of two- and three-dimensional isotropic porous materials can be characterised by two independent constants, the Young's modulus (E) and Poisson's ratio ($\nu$). In general, we expect the elastic constants to depend on properties of the solid matrix (which we denote with subscript s), or E = E_s f ( p, _s ) and = g( p, _s ). Here p is the relative density or solid volume fraction, and the form of the dimensionless functions f and g depend on microstructure. In two dimensions it has been observed that f and g have two remarkable properties for arbitrary porous materials (Day et al. , 1992). First, the Young's modulus is independent of $\nu _s$_s, or f (p,_s) = f (p). Second, if the solid fraction decreases to the percolation threshold p_c, the effective Poisson's ratio converges to a fixed point independent of the solid Poisson's ratio, or g (p, _s ) ₁ as p p_c. Both results were subsequently proved analytically by (Cherkaev, Lurie and Milton) (CLM) and Thorpe & Jasiuk (Thorpe & Jasiuk, 1992). (Christensen, 1993) has pointed out that although E cannot be independent of $\nu _s$_s in three-dimensions, it is nearly so for materials with dilute spheroidal voids over a restricted range of Poisson's ratio 0 _s 0.5. Variational bounds and approximate self-consistent theories showed a similar weak dependence of E on $\nu _s$_s (for 0 _s 0.5) over the full range of solid fraction 0 p 1. Since the CLM theorem, which is only true in 2-D, was used to prove the existence of the fixed point for $\nu$ in two dimensions (Thorpe & Jasiuk, 1992), this behaviour is also not thought to hold rigorously in three dimensions. However, it is interesting to examine how well these inherently two-dimensional results, both for E and for $\nu$, do approximately hold in three-dimensions.